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Hernalt
2011-May-30, 11:28 AM
:wall:

The blue chart on Wikipedia Delta-V (http://en.wikipedia.org/wiki/Delta-v) shows 2.5 km/s LEO to GTO and 0.7 GTO to Earth C3.

In trying to reproduce these two numbers, this is what I get:

v_{GTO Injection} - v_{LEO} = \sqrt{\mu_{earth}\(\frac{2}{r_{LEO}}-\frac{1}{a_{GTO}})}-\sqrt{\mu_{earth}\frac{1}{r_{LEO}}}

= 10.24 km/s - 7.78 km/s = 2.45 km/s

v_{GTOExitC3} - v_{GTOExit} = \sqrt{2}\sqrt{\mu_{earth}\(\frac{2}{r_{GEO}}-\frac{1}{a_{GTO}})}-\sqrt{\mu_{earth}\(\frac{2}{r_{GEO}}-\frac{1}{a_{GTO}})}

= 2.26 km/s - 1.60 km/s = .66 km/s

Am I doing that correctly? The progression implies you don't actually circularize orbit "into" GEO. I interpret that you just keep on trucking up "past" GEO.

(And / also / or) How do you do Earth C3 to Mars Transfer? Does Earth C3 by convention represent a particular orbital radius? It seems like it's suitable for any r using \sqrt{2} r. Coming out of a Hohmann transfer is - as far as I can tell for purposes of calculating escape velocity - exactly tangent to the circular orbit.

I did a calculation with \mu_{sun}, r_{earth orbit}, v_{earth orbit}, r_{mars orbit}, v_{mars orbit}, a_{earth mars hohmann} and got 5.5 km/s, which is much greater than the Wiki page's 0.6 km/s + 0.9 km/s. So I've got naive assumptions in there. It made sense to me that if you're outside the gravity well of both earth and mars, then neither \mu_{earth} or \mu_{mars} are used.

What obvious n00b fubars am I making? :doh: Thanks.

grapes
2011-May-30, 02:04 PM
(And / also / or) How do you do Earth C3 to Mars Transfer? Does Earth C3 by convention represent a particular orbital radius? It seems like it's suitable for any r using \sqrt{2} r. Coming out of a Hohmann transfer is - as far as I can tell for purposes of calculating escape velocity - exactly tangent to the circular orbit.Isn't C3 an escape "orbit"?

Hernalt
2011-May-30, 04:05 PM
As far as I can tell, yes. My reading of the chart on Wikipedia is that you take a GTO up to the radius of GEO, and then you have 3 options: 1) Let yourself fall back to LEO by not adding impulse. 2) Circularize yourself at the GEO radius by a particular impulse. 3) Put yourself on an escape orbit by an impulse to achieve \sqrt{2}v_{GEO}.

But at least from the wikipedia sequence, you're going immediately into a Mars transfer orbit, which must logically be half an ellipse. Whereas an escape orbit (zero energy) is supposed to be parabolic. :confused: I don't know when they actually teach space craft orbital mechanics, but I haven't seen it in the first two years of physics/astronomy. Perhaps it's only taught in aerospace?

grapes
2011-May-30, 04:21 PM
(And / also / or) How do you do Earth C3 to Mars Transfer? Does Earth C3 by convention represent a particular orbital radius? It seems like it's suitable for any r using \sqrt{2} r. Coming out of a Hohmann transfer is - as far as I can tell for purposes of calculating escape velocity - exactly tangent to the circular orbit.

I did a calculation with \mu_{sun}, r_{earth orbit}, v_{earth orbit}, r_{mars orbit}, v_{mars orbit}, a_{earth mars hohmann} and got 5.5 km/s, which is much greater than the Wiki page's 0.6 km/s + 0.9 km/s. So I've got naive assumptions in there. So, what r_{earth orbit} are you using?

Hernalt
2011-May-30, 04:39 PM
1 AU, and 1.5 AU for Mars (in meters of course)

grapes
2011-May-31, 01:31 PM
the Wiki page's 0.6 km/s + 0.9 km/s. Why do you add .6 and .9?

IsaacKuo
2011-May-31, 03:34 PM
As far as I can tell, yes. My reading of the chart on Wikipedia is that you take a GTO up to the radius of GEO, and then you have 3 options: 1) Let yourself fall back to LEO by not adding impulse. 2) Circularize yourself at the GEO radius by a particular impulse. 3) Put yourself on an escape orbit by an impulse to achieve \sqrt{2}v_{GEO}.
Options 1 and 2 are correct. Option 3 is not correct.

In order to maximize the benefit of the Oberth effect, you want to thrust at perigee rather than apogee. That means you go from LEO to GTO and you just keep on thrusting to Earth escape until you reach the desired C3.

This will put you on a hyperbolic escape trajectory; as it escapes the influence of Earth's gravity this trajectory should ideally match up with the desired Earth-Mars transfer orbit. (The patched conic method assumes a sharp cutoff between when the Earth's gravity is relevant and when the Sun's gravity is relevant. This approximation is good enough to make mission plans.)

What you have in mind is to first go from LEO to GTO, and then wait until you get to apogee before doing another thrust to get to Earth escape. This can work, but it will require much more delta-v than doing so at perigee. Because it's so inefficient, no one does it that way.

Anyway, look up "Oberth effect" to get some more understanding.

Hernalt
2011-May-31, 10:38 PM
Thank you, both, for pointers towards more research. I'm trying to learn about the handoff between spheres of influence (Earth soi => Sun soi => Mars soi) and the patching of conic sections, which are facets not revealed in my first googles. Oberth I've read about but I was only trying to reproduce the findings in the Wikipedia chart.

IsaacKuo
2011-Jun-01, 05:17 PM
In order to reproduce those findings, you need to assume a thrust at perigee rather than apogee. The Oberth effect is simply the principle which explains why perigee is the superior choice.